Is there a phase difference between current and voltage when AC electric field is applied across a resistor

Question

According to drude model when an alternating electric field is applied across a conductor, $$j = Re(\frac{ne^2\tau E}{m(1-i\omega\tau)})$$ where $E = E_0e^{-i\omega t}$. This suggests that the current and voltage across the conductor are not in phase, which violates the ohms law. Also while studying RLC circuits it is assumed that conductor(resistor) does not cause any phase change on its own, thus if an alternating field is applied only across resistor(there is no presence of capacitor or inductor in the circuit) then there should be no phase difference between current and applied voltage, but drude model predicts otherwise.

One reason to explain this which I thought of is- $$j = Re(\frac{ne^2\tau E}{m(1+\omega^2\tau^2)})-Re(i\frac{ne^2\omega\tau^2 E}{m(1+\omega^2\tau^2)})$$As $\tau$ is of order of $10^{-14}$ and $\omega $ of order of $10^{-1} $(generally) we can assume that $-Re(i\frac{ne^2\omega\tau^2 E}{m(1+\omega^2\tau^2)})$ is nearly zero, from this it can be concluded that there is a phase difference but it is negligible. Is this true?

I know that drude model has drawbacks but I did not saw prediction of phase difference as a drawback in any sources. Does there really exist a very small phase difference? Do other more accurate theory predict phase difference? Is phase difference ever observed experimentally?

Answer 1

First, note that circuit elements in circuit diagrams are generally showing idealizations that do not really exist in the real world. An ideal resistor is an object that exactly satisfies Ohm's law and therefore has zero reactance. In contrast, a real resistor (or any other electric component) will always have some capacitance (and inductance), however tiny it may be.

As for the Drude model, the reactance arises from the fact that model predicts that the current lags the electric field because it takes some time on the order of $\tau$ to react a change in the electric field. As you have indicated, at low frequencies, the electrons can keep up with the oscillations and the imaginary component is negligible. As frequency increases, the electrons have a harder time keeping up and the reactance grows. Above the plasma frequency, the electrons basically cannot keep up at all, the metal stops being reflective (the shininess arises from the fact that the electrons in the metal move to cancel out any incident electric field), and the metal essentially ceases to act like a resistor.